Flow Equation Approach to Periodically Driven Quantum Systems

We present a theoretical method to generate a highly accurate time-independent Hamiltonian governing the finite-time behavior of a time-periodic system. The method exploits infinitesimal unitary transformation steps, from which renormalization-group–like flow equations are derived to produce the effective Hamiltonian. Our tractable method has a range of validity reaching into frequency—and drive strength—regimes that are usually inaccessible via high-frequency $\omega$ expansions in the parameter $h/\omega$, where $h$ is the upper limit for the strength of local interactions. We demonstrate exact properties of our approach on a simple toy model and test an approximate version of it on both interacting and noninteracting many-body Hamiltonians, where it offers an improvement over the more well-known Magnus expansion and other high-frequency expansions. For the interacting models, we compare our approximate results to those found via exact diagonalization. While the approximation generally performs better globally than other high-frequency approximations, the improvement is especially pronounced in the regime of lower frequencies and strong external driving. This regime is of special interest because of its proximity to the resonant regime where the effect of a periodic drive is the most dramatic. Our results open a new route towards identifying novel nonequilibrium regimes and behaviors in driven quantum many-particle systems.

Popular Summary

One long-standing goal in physics is the ability to control the properties of quantum systems composed of many particles. This could allow researchers, for example, to tune a material to be insulating, superconducting, or something else entirely. A promising approach is to subject the system to periodically time-varying external fields, which can trigger phase transitions in the material. To predict and understand novel phases, researchers need effective mathematical models that accurately describe periodically driven systems. Here, we introduce a new method that extends current theoretical techniques into a new regime.

Much effort to date has focused on finding effective Hamiltonians in the analytically accessible high-frequency regime. However, this accessibility comes at a price: The necessary approximations are only valid in certain restricted situations. Going beyond these approximations is a long-standing problem, which hampers predictions of novel phases.

Our method provides a framework for developing Hamiltonians that work even when driving a system with a strong low-frequency external field. This advance means the low-frequency regime is more theoretically accessible than before. We apply our method to various quantum many-particle systems and compare the evolution of the system with the effective Hamiltonian obtained from our method and with the exact Hamiltonian to demonstrate its power.

We expect our method will help researchers map out the phase diagrams of various periodically driven quantum systems outside of the high-frequency regime, and we believe that it will provide a foundation for numerical and analytical approximation schemes.

Figure 1

The couplings as a function of flow parameter $s$ for the Hamiltonian in Eq. (19) with $B_p=3$, $B_z=1$, and $\omega =1$ in Eq. (17). This case corresponds to a low-frequency regime. Note that while the couplings exhibit a nontrivial dependence on $s$ until sufficiently large $s$, the unitary evolution remains stable down to small frequencies, as seen in the red curve (exact flow) in Fig. 3. The couplings after the range of the plot do not change within the limits of the line thicknesses.

Figure 2

The nonzero couplings as a function of frequency $\omega$ at the end of the flow (large $s$ values in Fig. 1) for $B_p=3$, $B_z=1$. Note that in spite of the rapid oscillations for small $\omega$, the resultant unitary evolution remains stable, as seen in the red curve (exact flow) in Fig. 3.

Figure 3

Plot of the $l_2$ distance between the time-evolution operator found by a Trotter expansion and the exact time-evolution operator obtained by exactly solving the flow equations in Eq. (6) given by Eq. (20) (red line), the Magnus expansion (blue line), and the rotating frame approximation given by Eq. (10) (black line).

Figure 4

Coupling constant $C_z^{+}$ as a function of flow parameter $s$ plotted for $\omega =1.2$.

Figure 5

Plot of the coupling constant $C_z^0$ as a function of $\omega$ for the flow equations solved up to a point $s=2000000$, which is long after the fixed point has been reached.

Figure 6

Plot of the $l_2$ distance between the exact time-evolution operator and the Magnus expansion (blue line), rotating frame approximation (orange line), and the solution of the exact flow equations but with a truncated ansatz (green line). The system size is $L=14$ sites.

Figure 7

Relative error, Eq. (42), of the time-evolution operator as a function of sampled $k$-points for the (a) Magnus expansion and (b) flow equation approach. Different driving frequencies $\omega =0.01$, $\omega =0.1$, $\omega =1$, and $\omega =3$ are considered. Note that the flow equation error is much smaller than the Magnus expansion error, particularly at the lowest frequencies. In both approximations, the error decreases as the frequency increases. We consider the case of $D=0.1$, $J_x=1$, $J_y=1.1$, $h_0=1$, and $h=1$.

Figure 8

Relative error $E$, Eq. (42), for $D=0.1$, $J_x=1$, $J_y=1.1$, $h_0=1$, and $h=1$ as a function of driving frequency $\omega$. Note how the flow equation approach outperforms the Magnus expansion, particularly at smaller $\omega$.

Figure 9

Relative error $E$, Eq. (42), as a function of the different coupling constants in the Hamiltonian, Eq. (32). Only one coefficient is varied in each subfigure, while the ones that are not varied are fixed at values of $D=0.1$, $\mathrm{\Delta }J=J_x-J_y=0.1$, $J=(J_x+J_y/2)=1.05$, $h_0=1$, and $h=1$. In panel (a), we vary the driving magnetic field $h$, in panel (b) the static magnetic field $h_0$, in panel (c) the average exchange interaction $J=(J_x+J_y/2)$, in panel (d) the antisymmetric exchange strength $D$, and in panel (e) the anisotropy of the exchange interaction $\mathrm{\Delta }J=J_x-J_y$.

Figure 10

Plot of the normalized $l_2$ distance between exact and approximate time-evolution operators for a chain of length $L=16$, and parameters $J_1=-0.5$, $J_1^z=1$, and $J_2=J_2^z=0$ plotted as a function of frequency for driving magnetic-field strengths (a) $B=0.5$ and (b) $B=5$.

Figure 11

Plots of the normalized $l_2$ distance for a chain of length $L=14$ where we vary the various coupling constants while keeping others fixed. Particularly, we are (a) varying nearest-neighbor exchange anisotropy $\mathrm{\Delta }J=J_1-J_1^z$ while keeping $B=1$, average nearest-neighbor exchange $J=(J_1+J_1^z/2)=1$, and $J_2=J_2^z=0$ fixed, (b) varying $B$ while keeping $J_1=-0.5$, $J_1^z=1$, and $J_2=J_2^z=0$ fixed, (c) varying $J$ while keeping $\mathrm{\Delta }J=1$, $B=1$, and $J_2=0$ fixed, and (d) varying $J_2$ while keeping $\mathrm{\Delta }J=1$, $B=1$, and $J=1$ fixed.

Figure 12

Plot of the normalized $l_2$ distance between exact and approximate time-evolution operators for $J_1=1$, $J_1^z=2$, and $J_2=0.2$ plotted as a function of frequency for a spin chain with $L=14$ sites. The driving strengths of the nearest-neighbor exchange term in the $z$ direction are (a) $J_{1,o}^z=2$ and (b) $J_{1,o}^z=6$.

Figure 13

Plots of the $l_2$ distance as a function of various coupling constants for a chain of length $L=14$. In panel (a), we vary the nearest-neighbor exchange anisotropy $\mathrm{\Delta }J=J_1-J_1^z$ and keep $J_{1,o}^z=2$, $J=\frac{1}{2}(J_1+J_1^z)=1.5$, and $J_2=0.2$ fixed; in panel (b), we vary the driving strength of the exchange interaction in the $z$ direction $J_{1,o}^z$ and keep $\mathrm{\Delta }J=-1$, $J=1.5$, and $J_2=0.2$ fixed; in panel (c), we vary the average nearest-neighbor exchange interaction $J$ and keep $J_{1,o}^z=2$, $\mathrm{\Delta }J=-1$, and $J_2=0.2$ fixed; and in panel (d), we vary the next-nearest-neighbor exchange $J_2$ while keeping $J_{1,o}^z=2$, $\mathrm{\Delta }J=-1$, and $J=1.5$ fixed.

Figure 14

Plot of the normalized $l_2$ distance between the exact time-evolution operator and the approximate time-evolution operator for $h_x=h\cos \theta$, $h_z=h\sin \theta$, and $\theta =0.643501$ using the “$\delta$ formulation” of the flow equations in orange and the “Heaviside $\theta$ formulation” in red. In plots (a)–(c), we keep $J_z=0.1$ fixed and plot different ranges of $h$ values, which are (a) short range $h$, (b) medium range $h$, and (c) long range $h$. Similarly, in plots (d)–(f), we keep $h=0.1$ fixed and plot different ranges of $J_z$ values, which are (d) short range $J_z$, (e) medium range $J_z$, and (f) long range $J_z$.